Render object twice with different materials
You can use multiple scenes:


Add a new scene by pressing the + at the top of the screen and selecting Link Objects:



This will link the objects into a new scene. This way modifying an object will change it in both scenes.
In the new scene, select the refractive object and pres U> Object. Now the refractive object in the new scene is no longer linked to the refractive object in the old scene, but the object data is still linked.

This mostly means that changes to the object transforms will not be copied between scenes, but changes to object data like materials and UV maps etc. will.
Still in the new scene, set the material to be stored in the object, not the object data:



Since the object is local to each scene, by setting the material to be stored in the object, we can now set the material individually for both scenes.

Remove the refractive material and add the reflective material. I made the reflective material slightly transparent with mask transparency so that it doesn't completely replace the refractive image they are combined, but you can do this in compositing too.
Still in the new scene, set the alpha to Transparent in Render settings > Shading, so the the sky is rendered as transparent alpha:



You only need to do this for the new scene.
In the old scene, add some composite nodes to combine the results of both scenes:



It doesn't really matter which scene you put the composite nodes in, but I like to keep everything in the "base" or "bottom" scene on top of which everything is overlaid. Just remember to render from the scene with the composite nodes.
Render. Both scenes should automatically render and combine.




Example .blend